Abstract
Now that we have seen the main consequences of the postulates of Special Relativity, i.e., the relativity of simultaneity, time dilation, and length contraction it is clear that the Galilei transformation, with its absolute time, is incorrect. These important physical phenomena can be seen as direct consequences of the correct transformation relating inertial frames, the Lorentz transformation. This transformation is the key for the formulation of Special Relativity in an enlightening four-dimensional formalism, which we will see in the next chapter. Here we study the Lorentz transformation and its properties and derive length contraction and time dilation directly from it, in addition to the transformation property of velocities. We must emphasize that, although the Lorentz transformation was discovered by studying the Maxwell equations, its validity is more general. The Lorentz transformation relates inertial frames without reference to the kind of physics studied in them. Lorentz-invariance is a general requirement for any physical theory, not just for electromagnetism.
Imagination is more important than knowledge.
—Albert Einstein.
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- 1.
Lorentz was also trying to explain the null result of the Michelson-Morley experiment by a physical contraction of the apparatus in the direction of motion. His interpretation, however, is rather misleading: the Lorentz transformation relates measurements performed in two different inertial systems.
- 2.
A priori, the constant \(c\) appearing in the Lorentz transformation is a fundamental velocity which needs not coincide with the speed of electromagnetic waves in vacuo, and is only later identified with it. This is not the historical route, in which the Lorentz transformation was derived from the Maxwell equations. There are many facets to the quantity \(c\) in various areas of physics (see Ref. [1] for a review).
- 3.
- 4.
- 5.
Our derivation of the Lorentz transformation from the postulates of Special Relativity requires only that the spatial part of the Lorentz transformation \(x'=G(v) \left( x-vt \right) \) reduces to the spatial part of the Galilei transformation\(x'=x-vt\) in the limit \( |v|/c \ll 1\), from which we deduced that \(G\rightarrow 1\). We did not assume that we recover \(t'=t\) in this limit, hence the proof is correct.
- 6.
If the particle traveling faster than light in that medium is charged, Cerenkov radiation is emitted.
- 7.
- 8.
This argument is due to E. F. Taylor [10].
- 9.
When introducing a set of axioms for a theory, one should always worry about the mutual consistency of these axioms. If two axioms are not consistent with each other, one is building an empty theory.
- 10.
This effect is the same phenomenon used advantageously in the optical lever of the torsion balance.
- 11.
A general wave, not necessarily electromagnetic, is discussed here. Also, quantum vacuum can behave as a medium and give rise to apparent velocities larger than \(c\). This is not, however, the propagation velocity of a physical signal and does not threaten causality [13].
- 12.
The phase velocity is usually identified with the velocity of a point of constant phase, for example a point where the amplitude of the wavepacket envelope vanishes.
- 13.
- 14.
Even group velocities can be larger than \(c\) if the wavepacket is too spread out in a medium with high absorption [16]. Again, we are not talking about physical velocities here.
- 15.
- 16.
This possibility will be applied to the derivation of the laws describing the aberration of light in Chap. 7
- 17.
The name “rapidity” arises from the one-to-one correspondence of \(\phi \) with the velocity \(v\) and the fact that \(\phi \approx \beta \) for \(|v| \ll c\).
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Faraoni, V. (2013). The Lorentz Transformation. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01107-3_2
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